Proposition 6.80. (Pro-left adjoint)
Let \(\phi_*\colon T \to S\) be a morphism of topoi. The induced functor
\[\Pro(\phi^*)\colon \Pro(S) \longrightarrow \Pro(T)\]
admits a left adjoint
\[\phi_\sharp\colon \Pro(T) \longrightarrow \Pro(S).\]
On corepresentable objects, it is characterized by
\[\phi_\sharp(X) = \Hom_T(X,\phi^*(-)) \qin \Fun^{\lex,\acc}(S,\An)\catop.\]
Its restriction \(\phi_\sharp\colon T\to\Pro(S)\) preserves colimits. Moreover, \(\phi_*\) is essential if and only if this restriction factors through \(S\hookrightarrow\Pro(S)\). In that case, the factorization is a left adjoint to \(\phi^*\).
Proof
Since \(\phi^*\) is accessible and left exact, the universal property of pro-categories gives the asserted left adjoint. For \(X\in T\) and \(Y\in S\), its adjunction isomorphism reads
\[\Hom_{\Pro(S)}(\phi_\sharp(X),Y)
\;\simeq\;
\Hom_{\Pro(T)}(X,\Pro(\phi^*)(Y))
\;\simeq\;
\Hom_T(X,\phi^*(Y)),\]
which gives the displayed formula. To see directly that the restriction to \(T\) preserves colimits, let \((X_i)_i\) be a diagram in \(T\). For every \(Y\in S\), we have \begin{align*} \Hom_{\Pro(S)}\bigl(\phi_\sharp(\colim_iX_i),Y\bigr) &\iso \Hom_T(\colim_iX_i,\phi^*Y) \\ &\iso \lim_i\Hom_T(X_i,\phi^*Y) \\ &\iso \Hom_{\Pro(S)}\bigl(\colim_i\phi_\sharp(X_i),Y\bigr). \end{align*} Since corepresentable objects detect isomorphisms in \(\Pro(S)\), the claim follows.Finally, the restriction factors through \(S\) if and only if every functor \(\Hom_T(X,\phi^*(-))\) is corepresentable by an object of \(S\). This says precisely that \(\phi^*\) admits a left adjoint.