That exactness conditions can be rephrased more explicitely as:
$$ Hom(V,Z) = \left\lbrace (v_i) \in \prod_i Hom(U_i,Z) \ \middle| \ \forall i,j,v_i \circ \pi_1 = v_j \circ \pi_2 \right\rbrace $$
where $\pi_1,\pi_2$ denotes the two projections $U_i \times_V U_j \rightrightarrows U_i,U_j$.
When you write it like this, the condition in the case $i=j$ is clearly vacuous when all the map $U_i \to U$ are monomorphism. Indeed in this case $U_i \times_V U_j$ is justs the intersection of $U_i$ and $U_j$, so that $\pi_1=\pi_2$ when $i = j$.
But in general it is not so. Let me consider the case where the you have a single map $U \to V$. Then the condition becomes
$$ Hom(V,Z) = \left\lbrace f \in Hom(U,Z) \ \middle| \ f\circ \pi_1 = f \circ \pi_2 \right\rbrace $$
where $\pi_1$ and $\pi_2$ are the two projections $U \times_V U \rightrightarrows U$.
You can think of $U \times_V U \rightrightarrows U$ as a map $U \times_V U \to U \times U$ which is a monomorphisms and corresponds to the "equivalence relation such that $V$ should be the quotient of $U$ by this relations".
So what you get is that a function from $V \to Z$ can be described as a function $U \to Z$ which is compatible to the equivalence relation $R$ such that $U/R \simeq V$.