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<p>'''Definition 1.3.'''  A ''functor'' <math>F : \mathcal{A} \to \mathcal{B}</math> is first of all a morphism of graphs (see Example C4), that is, it sends objects of <math>\mathcal{A}</math> to objects of <math>\mathcal{B}</math> and arrows of <math>\mathcal{A}</math> to arrows of <math>\mathcal{B}</math> such that, if <math>f : A \to A',</math> then <math>F(f) : F(A) \to F(A').</math>  Moreover, a functor preserves identities and composition;  thus:</p>

<p>'''Definition 1.3.'''  A ''functor'' <math>F : \mathcal{A} \to \mathcal{B}</math> is first of all a morphism of graphs &hellip;, that is, it sends objects of <math>\mathcal{A}</math> to objects of <math>\mathcal{B}</math> and arrows of <math>\mathcal{A}</math> to arrows of <math>\mathcal{B}</math> such that, if <math>f : A \to A',</math> then <math>F(f) : F(A) \to F(A').</math>  Moreover, a functor preserves identities and composition;  thus:</p>

<p>F(1_A) = 1_F(A),</p>

::<p><math>F(1_A) = 1_{F(A)}, \quad F(gf) = F(g)F(f).</math></p>

<p>F(gf)  =  F(g)F(f).</p>

<p>In particular, the identity functor <math>1_\mathcal{A} : \mathcal{A} \to \mathcal{A}</math> leaves objects and arrows unchanged and the composition of functors <math>F : \mathcal{A} \to \mathcal{B}</math> and <math>G : \mathcal{B} \to \mathcal{C}</math> is given by:</p>

<p>In particular, the identity functor 1_$A$ : $A$ -> $A$ leaves objects and arrows unchanged and the composition of functors F : $A$ -> $B$ and G : $B$ -> $C$ is given by:</p>

::<p><math>(GF)(A) = G(F(A)), \quad (GF)(f) = G(F(f)),</math></p>

<p>(GF)(A)  =  G(F(A)),</p>

<p>for all objects <math>A\!</math> of <math>\mathcal{A}</math> and all arrows <math>f : A \to A'</math> in <math>\mathcal{A}.</math></p>

 

<p>(GF)(f)  =  G(F(f)),</p>

 

<p>for all objects A of $A$ and all arrows f : A -> A' in $A$.</p>

<p>(Lambek & Scott, 6).</p>

<p>(Lambek & Scott, 6).</p>