F is a sheaf, iff $Hom(Y,F(-))$ is a sheaf

Question

So i have a problem with the following task:

Here is what I have tried so far: Let $U=\bigcup_{i\in I}U_i$ for opens $U_i$. We want to show that the following holds for each $a\in C$:

$Hom(a,F(U))\cong eq(\prod_{i\in I}Hom(a,F(U_i))\rightrightarrows \prod_{i,j\in I}Hom(a,F(U_i\cap U_j)))\cong eq(Hom(a,\prod_{i\in I}F(U_i))\rightrightarrows Hom(a,\prod_{i,j\in I}F(U_i\cap U_j)))$

I think i am almost done but i dont know how to continue here. I know that the Homfunctor preserves all limits and that the equalizer is one but not how to apply it here. Thank you!

Answer 1

You apply it directly: $\operatorname{eq}(\operatorname{Hom}(a,\prod_{i\in I}F(U_i))\rightrightarrows \operatorname{Hom}(a,\prod_{i,j\in I}F(U_i\cap U_j)))=$ $ = \operatorname{Hom}(a, \operatorname{eq}(\prod_{i\in I}F(U_i)\rightrightarrows \prod_{i,j\in I}F(U_i\cap U_j)).$

Since you assume that $F$ is a sheaf, the latter is $\operatorname{Hom}(a, F(U)),$ which is what you wanted.