Does the functor $-\times X:\mathbf{Set}\to\mathbf{Set}$ presrve limits?

Question

Let $X$ be a set. We know that the functor $$-\times X:\mathbf{Set}\to\mathbf{Set}$$ preserves all colimits because it is a left adjoint.

Does it preserve limits?

Answer 1

Short answer: no.

A little longer answer: consider the case $X=2$, the finite ordinal $\{0,1\}$, then we have that $$(2 \times 2) \times X=(2 \times 2) \times 2$$ has cardinality $8$ while $$(2 \times X) \times (2 \times X) \cong 2 \times 2 \times 2 \times 2$$ has cardinality $16$, hence they cannot be isomorphic.