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    $\begingroup$ Similar abuses of language include naming a model category by its fibrant objects ("the model category of quasicategories") or a 2-category by its 1-morphisms ("the 2-category of spans"). $\endgroup$ Commented Oct 24, 2009 at 22:03
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    $\begingroup$ yet nobody is brave enough to name categories from the name of arrows, like if we said "category of continuous mapping" for Top, etc. $\endgroup$ Commented Jun 25, 2010 at 10:55
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    $\begingroup$ @Pietro With the exception of Ehresmann and his school. :-) $\endgroup$ Commented Mar 13, 2011 at 15:24
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    $\begingroup$ I'd like to hear a convincing example where this has really been a problem. Usually there's a default notion of morphism (think of the category of sets, for instance), and in my experience, when anyone departs from the default, they make a point of it (e.g., the category or bicategory of sets and relations -- see, I didn't specify the 2-cells just now!). I hope Thierry can remember the details of his tale. $\endgroup$ Commented Aug 25, 2012 at 19:40
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    $\begingroup$ Ironically, I just had an example the other day (linear codes) where it wasn't completely clear to me what the correct notion of isomorphism should be!! So this is me answering my former (August 25 2012) self. $\endgroup$ Commented Apr 18, 2013 at 15:24