= Why is there no LOG in this?=C2=A0

It's=
a little messy to write out the actual details, but it is possible for eac=
h map f : X -> Y to use the subobject classifier and dependent products =
to construct a "universal lifting problem" for f, where universal=
means that every other lifting problem against a pushout product of monomo=
rphism and i-endpoint inclusion factors uniquely as a pullback followed by =
the universal lifting problem. Then, once a filler for the universal liftin=
g problem has been chosen, it can be used to construct a filler for all the=
other lifting problems. (This can also be seen as an instance of a general=
result that holds in any locally small fibration with finitely complete ba=
se).