That exactness conditions can be rephrased more explicitely as:

$$ Hom(V,Z) = \left\lbrace (v_i) \in \prod_i Hom(U_i,Z) \quad \middle| \quad \forall i,j, \ \ v_i|_{U_{i,j}} = v_j|_{U_{i,j}} \right\rbrace $$

where $U_{i,j} = U_i \times_U U_j$ and the vertical bar denote restriction (precomposition with the projection).

When you write it like this, the condition in the case $i=j$ is clearly vacuous, as the two terms are the same, so whether you include them or not in the last terms of the sequence do not change the anything.