of the form shown in the box at right.
Going either vertically or horizontally they form a category, with composition and identities.
Further, the compositions are compatible, a condition called the interchange law,
meaning that if four of them form a compatible 2x2 square, then
performing first horizontal, then vertical composition yields the same result as
performing first vertical, then horizontal composition.
That is a "symmetrical" definition of a double category.
There is also an unsymmetrical definition, which is also useful in many situations.
It defines a double category as an (internal) category in the category of categories, $\Cat$.
To best understand it we should briefly review the definition of internal category.
It starts with the notion of a graph in a category,
namely a diagram in that category of the form
\[\begin{array}{} X_0 & \mathop\leftleftarrows\limits^{\source s}_{\target t} & X_1\end{array}\]
Here $X_0$ is the "object of objects" and $X_1$ is the "object of arrows" of the category,
and the two arrows provide the source and target for the "arrows" in $X_1$.
A morphism of such graphs is a diagram of the form
\[\begin{array}{}
X_0 & \mathop\leftleftarrows\limits^{s_X}_{t_X} & X_1\\
\llap{f_0}\big\downarrow && \big\downarrow\rlap{f_1}\\
Y_0 & \mathop\leftleftarrows\limits^{s_Y}_{t_Y} & Y_1\\
\end{array}\]
which serially commutes, meaning that $s_X f_0 = f_1 s_Y$ and $t_X f_0 = f_1 t_Y$.
(If we view a graph $X$ in a category as
a functor $X$ from the abstract graph shape $\leftleftarrows$ to the category,
then this condition amounts to saying
$f$ is a natural transformation from the functor $X$ to the functor $Y$.)
Finally, to make a graph $X$ a category,
there must be defined a composition operation and an identity operation for the graph.
(We skip the details, see CWM, Chapter II, Section 7
or references on the web.)
Let use that definition to define an internal category in
the category of internal categories (internal to the category of small sets, $\Set$).
It has an underlying graph of the form (drawn vertically for reasons soon to appear)
\[\begin{array}{}
C_1\\
\llap\sv\source{\big\downarrow}\target{\big\downarrow}\rlap\tv\\
C_0 \end{array}\]
Since this is a diagram of internal categories,
$C_0$ and $C_1$ are themselves categories internal to $\Set$,
and $\sv$ and $\tv$ are internal functors, thus also graph morphisms, from $C_1$ to $C_0$.
Expanding the definitions, we have a diagram in $\Set$
\[\bbox[navajowhite,10px,border:4px groove red]{\begin{array}{}
V & = & (C_1)_0 & \mathop\leftleftarrows\limits^{\source{\textstyle\sh}}_{\target{\textstyle\th}} & (C_1)_1 & = & \{\,\text{squares}\,\}\\
& & \llap\sv\source{\big\downarrow} \target{\big\downarrow}\rlap\tv & & \llap\sv\source{\big\downarrow} \target{\big\downarrow}\rlap\tv \\
\{\,\text{0-cells}\,\} & = & (C_0)_0 & \mathop\leftleftarrows\limits^{\source{\textstyle\sh}}_{\target{\textstyle\th}} & (C_0)_1 & = & H\\
\end{array}}\]
which serially commutes
(i.e., when matching top and bottom, and left and right, arrows are paired;
there are four such pairs, to be depicted momentarily).
The reasons for the $V$ and $H$ will also appear momentarily.
Okay, that diagram gives the internal category representation of this structure.
But now let us redraw that same data in a form which shows how this gives
the source and target operations for squares,
and explicitly shows the four squares (in $\Set$) required to commute to give serial commutativity.
Specifically:
\[\bbox[navajowhite,10px,border:4px groove red]{\begin{array}{}
\\
&&& H\\[-4ex]
&&& \parallel\\[-4ex]
& (C_0)_0 & \source{\xleftarrow{\textstyle\sh}} & (C_0)_1 & \target{\xrightarrow{\textstyle\th}} & (C_0)_0\\
& \llap{\textstyle\sv}\source{\big\uparrow} && \llap{\textstyle\sv}\source{\big\uparrow} && \source{\big\uparrow}\rlap{\textstyle\sv}\\
V =\!\!\! & (C_1)_0 & \source{\xleftarrow{\textstyle\sh}} & (C_1)_1 & \target{\xrightarrow{\textstyle\th}} & (C_1)_0 & \!\!\!= V\\
& \llap{\textstyle\tv}\target{\big\downarrow} && \llap{\textstyle\tv}\target{\big\downarrow} && \target{\big\downarrow}\rlap{\textstyle\tv}\\
& (C_0)_0 & \source{\xleftarrow[\textstyle\sh]{}} & (C_0)_1 & \target{\xrightarrow[\textstyle\th]{}} & (C_0)_0\\[-2ex]
&&& \Vert\\[-2ex]
&&& H\\
\end{array}}\]
So we have:
$(C_1)_1$ = the set of all the squares,
$(C_1)_0$ = the set of all the vertical arrows, which we may call $V$,
$(C_0)_1$ = the set of all the horizontal arrows, which we may call $H$, and
$(C_0)_0$ = the set of all the 0-cells,
and $\sv$, $\tv$, $\sh$, and $\th$ give the source-target operations at all the requisite levels.
The sketch for the last display:
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