What category are Tagless Final Algebras final In?
Final Algebra semantics was introduced by Mitch Wand in his paper "Final Algebra Semantics and Data Type Extensions", see this freely available tech report: https://www.cs.indiana.edu/ftp/techreports/TR65.pdf .

It does not mean final coalgebra semantics, which is a very different idea. The wrinkle is that the algebra is not final in the same category that the initial algebra is initial, because then the final algebra would just be given by the unit type/terminal object as carrier. Instead, Wand defines a category of adequate extensions of an algebra, which basically say you want the algebras in question to not equate basic values. An example of this is that the signature may contain numbers and you only want to consider algebras where the interpretations of distinct integers are unequal. This rules out the trivial terminal object as carrier because it would equate all numbers. I wrote a blog post about this a while ago in an attempt to understand the same question: http://prl.ccs.neu.edu/blog/2017/09/27/final-algebra-semantics-is-observational-equivalence/ .