Pushouts: uniting and identifying

C / \ f / \ g / \ |_ _| A B \ / \ / \ / _| |_ A +_C B

Given a span of sets as shown above
(i.e, sets A, B, C and functions f, g between them as shown above),
we "add", that is, take the disjoint union of, the sets A and B,
then, for each element c in C,
identify f(c) (in the union) with g(c) (in the union).
(Actually, we take the smallest equivalence relation which includes all such identifications,
and divide the A + B by that equivalence relation.)
The resulting cospan diagram (depicted above>
is called the pushout
of the span diagram determined by f and g.
As just defined,
it evidently combines both uniting (taking the disjoint union of) A and B
with identifying elements of A and B as determined by the span with vertex C.