Lemma 2.8.
For a category \(T\) with pullbacks and \(I\)-indexed colimits, the following conditions are equivalent:
\(I\)-indexed colimits are universal in \(T\);
For every diagram \(X_{\bullet}\colon I \to T\), the counit of the adjunction \(T_{/X} \leftrightarrows \lim_i T_{/X_i}\) is an isomorphism;
For every diagram \(X_{\bullet}\colon I \to T\), the functor \(T_{/X} \to \lim_i T_{/X_i}\) is fully faithful.
Proof
The equivalence between (2) and (3) is the standard characterization of a fully faithful right adjoint. If colimits are universal, then for every \(Y \to X=\colim_iX_i\) we have
\[\colim_i(X_i\times_XY) \iso X\times_XY \iso Y,\]
so the counit is an isomorphism.Conversely, assume (2), let \(A \to B\) be a morphism, and let \(X_{\bullet}\) be an \(I\)-indexed diagram in \(T_{/B}\) with colimit \(X \to B\). Apply (2) to the underlying diagram \(X_{\bullet}\) and the object \(A\times_BX \to X\). The resulting counit is \[\colim_i\bigl(X_i\times_X(A\times_BX)\bigr) \iso A\times_BX.\]
Since \(X_i\times_X(A\times_BX)\iso A\times_BX_i\), this says exactly that pullback along \(A\to B\) preserves the given colimit. Hence \(I\)-indexed colimits are universal.