I'm learning category theory. There's a homework question asking for a pushout of groupoid. Suppose $C_0,C_1,C_2$ are groupoids and denote $f_i:C_0\to C_i\quad i=1,2$ the functor.
I have managed to show the objects are $\mathrm{Obj}(C_1 \underset{C_0}{\sqcup}C_2 )=(C_0/\sim)\cup (C_1\backslash f_1^{-1}(C_0))\cup (C_2\backslash f_2^{-1}(C_0))$ where the equivalence is the "closure" of the relation having same images in $C_1$ or $C_2$.
I think that the morphisms can be generated by the morphisms above, but have problems making it clear.