>First use \$$f: a' \to a\$$ to cook up a map

>$\mathcal{B}(F(a),b) \to \mathcal{B}(F(a'),b) .$

What is happening to cause a morphism,

\$f: a' \to a\$

to be flipped backward when being mapped to,

$\mathcal{B}(F(f),b) : \mathcal{B}(F(a),b) \to \mathcal{B}(F(a'),b) ?$