I was reading the second chapter of Sheaves in geometry and logic by Mac Lane and there he says the following,
On a space X, each open set U determines a presheaf Hom(-,U) defined, for each open set V, by Hom(V,U)=1 if $V \subset U$ or $Hom(V,U)= \emptyset$ otherwise.
They say that this sheaf is clearly a sheaf and I was wondering, in order to prove the two sheaf axioms, how I would build the respective equalizer. Just a note, my definition of a sheaf (also given by this book) is the following:
A sheaf of sets F on a topological space X is a functor $F:\mathscr{O}(X)^{op} \mapsto Sets$ such that each open covering $U=\cup_iU_i$, $i\in I$ of an open set U of X yields an equalizer diagram
I saw some people here use something as the sheaf Hom (like here: prove-that-sheaf-hom-is-a-sheaf) and i was wondering if it was the same thing that I am trying to build.
Thank you for your help in advance!
Your sheaf and the sheaf mentioned in the linked question are two different objects.
Your sheaf is the representable presheaf corresponding to the open $U$, meaning that it maps an object $V$ to the set of morphisms $V\rightarrow U$, which is a one-point set if $V\subseteq U$ and the empty set otherwise.
In the linked question, they are defining the sheaf of morphisms $\mathcal F\rightarrow\mathcal G$ between two sheaves $\mathcal F,\mathcal G$. This sheaf has the property that global sections are exactly the set of morphisms of sheaves $\mathcal F\rightarrow\mathcal G$ (i.e. natural transformations between functor $\mathscr O(X)^{op}\rightarrow\mathcal Set$). On the other hand, the global sections of $\text{Hom}(-,U)$ are empty if $U\neq X$ and are a one-point set if $U=X$.
You can see the difference also in the inputs. Your sheaf takes as input a single open set $U$ and constructs a sheaf out of it. In the linked question, they take as input two sheaves $\mathcal F,\mathcal G$.