For any commutative ring R, we show that the categories ofR-coalgebras and cocommutative R-coalgebras are locally \(\aleph_1\) -presentable, while the categories of R-flatR-coalgebras are \(\aleph_1\) -accessible. Similarly, for any associative ring R, the category of R-coringsis locally \(\aleph_1\) -presentable, while the category ofR-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 abstractcategory-theoretic principle going back to Ulmer. For right or two-sided R-module flat R-corings, our cardinalityestimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categoriesis also considered.
