Functoriality of functor categories

Construction 1.3.20 (Functoriality of functor categories in \(C\) and \(D\)). Given a functor \(g\colon D \to E\), we define the functor \(g \circ -\colon \Fun (C,D) \to \Fun (C,E)\) as the currying of the composite \[ \Fun (C, D) \times C \xrightarrow {\ev } D \xrightarrow {g} E. \] Similarly, given a functor \(f\colon C \to D\), we define the functor \(- \circ f\colon \Fun (D,E) \to \Fun (C,E)\) as the currying of the composite \[ \Fun (D, E) \times C \xrightarrow {\id \times f} \Fun (D,E) \times D \xrightarrow {\ev } E. \] In a completely analogous way, every natural isomorphism \(\beta \colon g \cong g'\) of functors \(D \to E\) induces a natural isomorphism \((\beta \circ -) \colon (g \circ -) \cong (g' \circ -)\) of functors \(\Fun (C,D) \to \Fun (C, E)\), and similarly for the construction \(- \circ f\).

Alternative notations for these functors that we will frequently use are \(g_*\) and \(f^*\): \[ g_* := g \circ -\colon \Fun (C,D) \to \Fun (C,E), \qquad f^*:= - \circ f\colon \Fun (D,E) \to \Fun (C,D). \]