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"?