The category of graphs over a set X

We show a cube of adjunctions relating
the category of graphs over a given fixed set $X$
to seven other related categories.

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A note on terminology:
We use the word "graph" in the sense it is often used in category theory,
in, for example, Mac Lane's Categories for the Working Mathematician,
to denote a parallel pair of functions between two sets, e.g., $E \rightrightarrows V$, or more generally,
to denote a parallel pair of arrows in any category.
Such a graph in a category may be identified with a functor from the abstract parallel pair $\rightrightarrows$ to the category.
The notion of morphism of graphs then is just a natural transformation between such functors: \[\begin{array}{} E_0 & \mathop\leftleftarrows\limits^{s_E}_{t_E} & E_1\\ \llap{f_0}\big\downarrow && \big\downarrow\rlap{f_1}\\ F_0 & \mathop\leftleftarrows\limits^{s_F}_{t_F} & F_1\\ \end{array}\] In this document we consider graphs, in that sense, in the category of small sets, $\Set$,
which are the objects of a category denoted $\Graph$.

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We restrict our attention further to graphs having a fixed small set $X$ as their set of vertices,
and graph morphisms between such graphs which are the identity function $1_X$ on vertices,
thus graph morphisms of the form \[\begin{array}{} X & \mathop\leftleftarrows\limits^{s_E}_{t_E} & E\\ \llap{1_X}\big\downarrow && \big\downarrow\rlap h\\ X & \mathop\leftleftarrows\limits^{s_F}_{t_F} & F\\ \end{array}\] normally simplified by replacing $1_X$ with an $=$ sign.
The category of such graphs over $X$ is here for brevity denoted $\Graph_X$.
We consider a number of related categories having $X$ as their set of vertices, or objects.
To denote this restricted notion,
we apply a suffix ${}_X$ to the notation for the unrestricted notion,
thus, for example, $\Cat_X$ denotes
the category of categories having $X$ as their set of objects, and functors between such categores that are the identity on $X$.

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We then have the following cube of (large) categories and adjunctions between them: \[\bbox[20px,border:4px groove red]{\begin{array}{} \leftcat{\Graph_X} & \leftadj{\xrightarrow{\displaystyle\text{gen}}} \atop \rightadj{\xleftarrow[\displaystyle\text{forget}]{}} & \Cat_X\\ & \leftadj{\llap{\text{gen}}\searrow} \rightadj{\nwarrow\rlap{\text{forget}}} && \leftadj{\llap{\text{gen}}\searrow} \rightadj{\nwarrow\rlap{\text{forget}}}\\ && \SymGraph_X & \leftadj{\xrightarrow{\displaystyle\text{gen}}} \atop \rightadj{\xleftarrow[\displaystyle\text{forget}]{}} & \Groupoid_X\\ \leftadj{\llap{\text{image} \atop \text{support}}\Bigg\downarrow} \rightadj{\Bigg\uparrow\rlap{\text{include}}} &&&& \leftadj{\llap{\text{image} \atop \text{support}}\Bigg\downarrow} \rightadj{\Bigg\uparrow\rlap{\text{include}}}\\ \Reln_X & \leftadj{\xrightarrow{\displaystyle\text{gen}}} \atop \rightadj{\xleftarrow[\displaystyle\text{forget}]{}} & \PreOrd_X\\ & \leftadj{\llap{\text{gen}}\searrow} \rightadj{\nwarrow\rlap{\text{forget}}} && \leftadj{\llap{\text{gen}}\searrow} \rightadj{\nwarrow\rlap{\text{forget}}}\\ && \SymReln_X & \leftadj{\xrightarrow{\displaystyle\text{gen}}} \atop \rightadj{\xleftarrow[\displaystyle\text{forget}]{}} & \rightcat{\EquivReln_X}\\ \end{array}}\] Here the back right and front left vertical edges were omitted
due to the limitations of the rectilinear graphics.
Those omitted vertical edges are: \[\begin{array}{} \Cat_X\\ \leftadj{\llap{\text{image} \atop \text{support}}\Bigg\downarrow} \rightadj{\Bigg\uparrow\rlap{\text{include}}}\\ \PreOrd_X \end{array} \qquad\quad\text{and}\qquad \begin{array}{} \SymGraph_X\\ \leftadj{\llap{\text{image} \atop \text{support}}\Bigg\downarrow} \rightadj{\Bigg\uparrow\rlap{\text{include}}}\\ \SymReln_X \end{array}\]

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The above diagram was an attempt to show the cube of adjunctions
in three-dimensional perspective.
Here is a different presentation of the cube, ${\text{Up} \atop \text{Front}\; \text{Right}} = {\text{Left}\; \text{Back} \atop \text{Down}}$
where on either side of the equals sign appear
three faces of the cube, flattened into a plane.
Only the left adjoint functors are shown. \[\bbox[20px,border:4px groove red]{\begin{array}{} 3& && \leftcat{\Graph_X} && && && \leftcat{\Graph_X}\\ & & \leftadj\swarrow && \leftadj\searrow & && & \leftadj\swarrow & \leftadj\downarrow & \leftadj\searrow\\ 2& \SymGraph_X &&&& \Cat_X && \SymGraph_X && \Reln_X && \Cat_X\\ & \leftadj\downarrow & \leftadj\searrow && \leftadj\swarrow & \leftadj\downarrow & \qquad = \qquad & \leftadj\downarrow & \leftadj\swarrow && \leftadj\searrow & \leftadj\downarrow\\ 1& \SymReln_X && \Groupoid_X && \PreOrd_X && \SymReln_X &&&& \PreOrd_X\\ & & \leftadj\searrow & \leftadj\downarrow & \leftadj\swarrow & && & \leftadj\searrow & & \leftadj\swarrow\\ 0& && \rightcat{\EquivReln_X} && && && \rightcat{\EquivReln_X} \\ \end{array}}\]