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See pages 120-123 of MacLane's "Categories for the working mathematician". The local  (co)boundedness condition has actual content. For example:

See pages 120-123 of MacLane's "Categories for the working mathematician". 

The local (co)boundedness condition has actual content. For example:

b) Functors ${\bf Group}\rightarrow {\bf Set}$, continuous but admitting no left adjoint, may be obtained as follows: let $\Gamma_\alpha$ be a simple group of cardinality $\aleph_\alpha$ (e.g. the alternating group on a set of that cardinality, or the projective special linear group on a 2-dimensional vector space over a field of that cardinality) and take the product (suitably construed), over the proper class of all ordinals, of the functors ${\rm Hom}(\Gamma_\alpha,-)$.

b) Functors ${\bf Group}\rightarrow {\bf Set}$, continuous but admitting no left adjoint, may be obtained as follows: let $\Gamma_\alpha$ be a simple group of cardinality $\aleph_\alpha$ (e.g. the alternating group on a set of that cardinality, or the projective special linear group on a 2-dimensional vector space over a field of that cardinality) and take the product (suitably construed), over the proper class of all ordinals, of the functors ${\rm Hom}_{\bf Group}(\Gamma_\alpha,-)$.

A) Some colleagues of mine once made the following disclaimer: 'The "set" of stable curves does not exist, but we leave this set theoretic difficulty to the reader.' The colleagues (names withheld to protect the innocent) are of course fully aware of the fact that strictly speaking, the class of all stable curves, or topological spaces, or groups, or any the other usual suspects customarily formalized in terms of structured sets, cannot itself be a set. While they recognize that the structure they study is transportable along arbitrary bijections between members of a proper class of equinumerous sets, they also recognize that in their setting this same transportability could justify a technically sufficient a priori restriction to some fixed but otherwise arbitrary underlying set: that is, the relevant large category has a small skeletal subcategory. (Exercise: precisely what makes this work in the example given?) In such cases, the set versus class pecadillo is an essentially victimless one, perhaps barring discussions of the admissibility of Choice, expecially Global Choice.

A) Some colleagues of mine once made the following disclaimer: 'The "set" of stable curves does not exist, but we leave this set theoretic difficulty to the reader.' These colleagues (names withheld to protect the innocent) are of course fully aware of the fact that strictly speaking, the class of all stable curves, or topological spaces, or groups, or any the other usual suspects customarily formalized in terms of structured sets, cannot itself be a set. While they recognize that the structure they study is transportable along arbitrary bijections between members of a proper class of equinumerous sets, they also recognize that in their setting this same transportability could justify a technically sufficient a priori restriction to some fixed but otherwise arbitrary underlying set: that is, the relevant large category has a small skeletal subcategory. (Exercise: precisely what makes this work in the example given?) In such cases, the set versus class pecadillo is an essentially victimless one, perhaps barring discussions of the admissibility of Choice, expecially Global Choice.