The category OS of operator spaces and complete contractions is locally countably presentable, and cofree (cocommutative) coalgebras exist for the projective tensor product.
Handbook of Categorical Algebra: Volume 2, Categories and Structures , volume 2
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The Category of Operator Spaces and Complete Contractions
The category OS of operator spaces and complete contractions is locally countably presentable, and cofree (cocommutative) coalgebras exist for the projective tensor product.