If F is a functor from $C_{1} \times C_{2}$ does this equlity hold: $F(f, h \circ g) =F(f,h) \circ F(f,g)$?

Question

Lets say we have a functor $F:C_{1} \times C_{2} \rightarrow D$, where $C_{1},C_{2},D$ are catagories. Does this equation hold for funtors: $F(f, h \circ g) =F(f,h) \circ F(f,g)$, where $f,h,g$ are morpshism?

Answer 1

I don't think this makes sense. If you have $F(f,h\circ g)$ you need to have an arrow $f:A_0\to A_1$ in the category $C_1$ and arrows $g:B_0\to B_1$ and $h:B_1\to B_2$ in the category $C_2$. Then $F(f,g)$ is an arrow from $F(A_0,B_0)$ to $F(A_1,B_1)$ in $D$ and $F(f,h)$ is an arrow from $F(A_0,B_1)$ to $F(A_1,B_2)$ in $D$. In general $F(A_1,B_1)\ne F(A_0,B_1)$ so you can't compose these arrows.

But you do have $$F(f,h\circ g)=F(f,h)\circ F(\text{id}_{A_0},g)$$ and $$F(f,h\circ g)=F(\text{id}_{A_1},h)\circ F(f,g).$$