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Exercise In the situation of Definition, show that we can replace $(a)$ and $(b)$ by the following alternative conditions:

  • For every object $X \in \operatorname{\mathcal{C}}$, the diagram

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{id}_{F(X)} \circ \operatorname{id}_{ F(X)} \ar@ {=>}[r]^-{ \upsilon _{F(X)} } \ar@ {=>}[d]^{ \epsilon _{X} \circ \epsilon _{X}} & \operatorname{id}_{ F(X) } \ar@ {=>}[dd]^{ \epsilon _{X} } \\ F( \operatorname{id}_ X) \circ F( \operatorname{id}_ X ) \ar@ {=>}[d]^{ \mu _{ \operatorname{id}_ X, \operatorname{id}_ X } } & \\ F( \operatorname{id}_ X \circ \operatorname{id}_ X ) \ar@ {=>}[r]^{ F( \upsilon _{X} )} & F(\operatorname{id}_ X) } \]

    commutes (in the endomorphism category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$).

  • For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the vertical compositions

    \[ \operatorname{id}_{F(Y)} \circ F(f) \xRightarrow { \epsilon _{Y} \circ \operatorname{id}_{F(f)} } F( \operatorname{id}_{Y} ) \circ F(f) \xRightarrow { \mu _{ \operatorname{id}_ Y, f} } F( \operatorname{id}_ Y \circ f) \]
    \[ F(f) \circ \operatorname{id}_{F(X)} \xRightarrow { \operatorname{id}_{F(f)} \circ \epsilon _ X } F(f) \circ F( \operatorname{id}_ X) \xRightarrow { \mu _{ f, \operatorname{id}_ X} } F(f \circ \operatorname{id}_ X) \]

    are monomorphisms in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$.

See Proposition