Covariant functor

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Definition

A covariant functor, $T:C\leadsto D$ (for categories $C$ and $D$ ) is a pair of mappings^[1]:

$T:\left\{\begin{array}{rcl}\text{Obj}(C) & \longrightarrow & \text{Obj}(D)\\ X & \longmapsto & TX \end{array}\right.$
$T:\left\{\begin{array}{rcl}\text{Mor}(C) & \longrightarrow & \text{Mor}(D)\\ f & \longmapsto & Tf \end{array}\right.$

Which preserve composition of morphisms and the identity morphism of each object, that is to say:

\forall f,g\in\text{Mor}(C)[Tfg=T(f\circ g)=Tf\circ Tg=TfTg] (I've added the \circs in to make it more obvious to the reader what is going on)
- Where such composition makes sense. That is $\text{target}(g)=\text{source}(f)$ .
and $\forall A\in\text{Obj}(C)[T1_A=1_{TA}]$

Thus if $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are morphisms of $C$ , then the following diagram commutes:

$\begin{xy}\xymatrix{TX \ar[rr]^{Tgf} \ar[dr]_{Tf} & & TZ \\ & TY \ar[ur]_{Tg} & }\end{xy}$ Thus the diagram just depicts the requirement that: $=Tgf=Tg\circ Tf$	$\$	Note that the diagram is basically just the "image" of $\begin{xy}\xymatrix{X \ar[rr]^{gf} \ar[dr]_{f} & & Z \\ & Y \ar[ur]_{g} & }\end{xy}$ under $T$

Given

3 objects, $X$ , $Y$ and $Z$ in a category $\mathscr{C}$
a (covariant) functor from \mathscr{C} to another category, \mathscr{D}
- $T:\mathscr{C}\leadsto\mathscr{D}$
morphisms $f:X\rightarrow Y$ , $g:Y\rightarrow Z$ and the morphism $gf:X\rightarrow Z$ corresponding to the composition $g\circ f$

The functor gives us "the same" diagram (in terms of objects and arrows) in the target category $\mathscr{D}$ , as shown by the following diagram:

The dashed lines represent $T$ 's image of objects The dotted lines are the image of morphisms under $T$
$\xymatrix{ X \ar@{-->}@(u,ul)[rrrr] \ar[rr]^{gf}="gf" \ar[dr]_f="f" \ar& & Z \ar@{-->}@(u,ul)[rrrr] & & TX \ar[rr]^{Tgf}="tgf" \ar[dr]_{Tf}="tf" & & TZ\\ & Y \ar[ur]_{g}="g" \ar@{-->}@(d,dl)[rrrr] & & & & TY \ar[ur]_{Tg}="tg" \ar@{.>}@/^/ "gf";"tgf" \ar@{.>}@/_/ "f";"tf" \ar@{.>}@/^/ "g";"tg" }$