Free presheaf monad [mde-J7Q3]

For a small category \(\mathbb {C}\), the free presheaf monad is given by \[ \begin {align} \mathsf {Psh}_{\mathbb {C}} : \mathbf {Set}^{\mathbb {C}_{\text {obj}}} &\to \mathbf {Set}^{\mathbb {C}_{\text {obj}}} \\ \{X_c\}_{c \in \mathbb {C}} &\mapsto \left \{ \coprod _{c' \in \mathbb {C}} X_{c'} \times \mathbb {C}(c,c') \right \}_{c \in \mathbb {C}} \end {align} \] with unit \[ \begin {align} {\eta _X}_c : X_c &\to \coprod _{c' \in \mathbb {C}} (X_c' \times \mathbb {C}(c,c')) \\ x &\mapsto (c, x, \textsf {id}_{c}) \end {align} \] and multiplication \[ \begin {align} {\mu _X}_c : \coprod _{e \in \mathbb {C}} \coprod _{d \in \mathbb {C}} X_d \times \mathbb {C}(e,d) \times \mathbb {C}(c,e) &\to \coprod _{d \in \mathbb {C}} X_d \times \mathbb {C}(c,d)\\ (e,d,x,f,g) &\mapsto (d,x,f \circ g). \end {align} \]