Keith wrote:

> What is happening to cause a morphism,

> \$f: a' \to a\$

> to be flipped backward when being mapped to,

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

Peter Addor gave the formula for this map. It sends any guy

$h \in \mathcal{B}(F(a),b)$

to

$h \circ F(f) \in \mathcal{B}(F(a'),b) .$

That is, it sends any guy

$h: F(a) \to b$

to

$h \circ F(f) : F(a') to b .$

We are _precomposing_ with \$$F(f)\$$.

If you think about this, you'll see why we need \$$f: a' \to a\$$, not \$$f: a \to a'\$$.

_Postcomposing_ does not cause this flip.

Simply put: if you're riding from Philadelphia to New York, and you want to extend your trip, you can either postcompose with a trip from New York to Boston, or precompose with a trip from Washington DC to Philadelphia.

The second option, stretching out our trip so it includes Washington DC, does not require a trip from your original starting point (Philadelphia) to your new one (Washington DC). It requires a trip _the other way_, from Washington DC to Philadelphia! This is the 'flip'.

It doesn't feel like a flip when you put it this way; it seems utterly reasonable.