categories: adjoining an indeterminate

[jdolan@math.ucr.edu]

in lambek and scott's "introduction to higher order categorical
logic", in the "historical comments" on section 7 of part 1, on page
116, they say:

"it should be emphasized that, as long as equalizers are excluded from
the definition of cartesian closed categories, adjoining an
indeterminate of type a is not the same as forming the slice category
c/a, but it is once equalizers are included.  the latter was observed
by grothendieck and joyal (see part 2, section 16, exercise 2)."

i've been having a bit of trouble trying to understand exactly what
they mean here.  as far as i can tell, the mentioned "exercise 2"
concerns the case where the category c is a topos, but the above
quoted statement makes it sound like the equivalence between adjoining
an indeterminate and forming a slice category should hold in the case
where c is just a "cartesian closed category with equalizers", or
something like that.  offhand though i couldn't think of a way to get
a result along these lines to be true.  for one thing, the assumption
that c has finite limits and exponentials doesn't even seem to compel
the slice category c/a to have exponentials.  (i think the category of
co-commutative co-algebras over a field provides a counterexample.)

can someone explain where i'm making a mistake here?  or is it just
that lambek and scott were only referring to the topos case, as
discussed in their "exercise 2"?