lemma. adjunction preserves (co)limits [leinster2016basic, 6.3.1] [tt-002O]

✍️source

Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), \(\mathscr {L}\) preserves colimits, and \(\mathscr {R}\) preserves limits.

Explictly, given \(\mathscr {D}: {\cal J} \to {\cal C}\), we have \[(\operatorname {colim} \mathscr {D}) \mathbin {\bullet } \mathscr {L} \cong \operatorname {colim}(\mathscr {D} \mathbin {\bullet } \mathscr {L})\] and given \(\mathscr {D}': {\cal J} \to {\cal D}\), we have \[(\lim \mathscr {D}') \mathbin {\bullet } \mathscr {R} \cong \lim (\mathscr {D}' \mathbin {\bullet } \mathscr {R})\]