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For any commutative ring $R$, we show that the categories of $R$-coalgebrasand cocommutative $R$-coalgebras are locally $\aleph_1$-presentable, while thecategories of $R$-flat $R$-coalgebras are $\aleph_1$-accessible. Similarly, forany associative ring $R$, the category of $R$-corings is locally$\aleph_1$-presentable, while the category of $R$-$R$-bimodule flat $R$-coringsis $\aleph_1$-accessible. The cardinality of the ring $R$ can be arbitrarilylarge. We also discuss $R$-corings with surjective counit and flat kernel. Theproofs are straightforward applications of an abstract category-theoreticprinciple going back to Ulmer. For right or two-sided $R$-module flat$R$-corings, our cardinality estimate for the accessibility rank is not asgood.

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