The categories of corings and coalgebras over a ring are locally countably presentable

For any commutative ring R, we show that the categories of R-coalgebras and cocommutative R-coalgebras are locally \(\aleph_1\)-presentable, while the categories of R-flat R-coalgebras are \(\aleph_1\)-accessible. Similarly, for any associative ring R, the category of R-corings is locally \(\aleph_1\)-presentable, while the category of R-R-bimodule flat R-corings is \(\aleph_1\)-accessible. The cardinality of the ring R can be arbitrarily large. We also discuss R-corings with surjective counit and flat kernel. The proofs are straightforward applications of an abstract category-theoretic principle going back to Ulmer. For right or two-sided R-module flat R-corings, our cardinality estimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categories is also considered.