For posets $A,B$, let $B^A$ be the poset of monotone maps $A\to B$ ordered pointwise. Evaluation
$$
B^A\times A\to B,\qquad(f,a)\mapsto f(a)
$$
is monotone. A monotone map $h:C\times A\to B$ curries to the monotone map
$$
\widehat h:C\to B^A,\qquad
\widehat h(c)(a)=h(c,a),
$$
and this correspondence is natural and invertible. Thus the <cartesian closed category of posets> has exponentials $B^A$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 5 / b / Solution

The exponential $B^A$ in the category of posets is the poset of monotone maps $A\to B$ ordered pointwise. Evaluation is monotone, and currying gives the cartesian-closed adjunction.


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 Incoming links: Cartesian closed category of posets