We show how the relatively initial or relatively terminal fixed points for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find sufficient conditions for existence of this adjunction. We show that relative fixed points may be characterized as (co)equalizers of the free (co)monad on $F$. In particular, when $F$ is a polynomial functor on $\mathsf{Set}$ the relative fixed points are a quotient or subset of the free term algebra or the cofree term coalgebra. We give examples of the relative fixed points for polynomial functors and an example which is the Sierpinski carpet. Lastly, we prove a general preservation result for relative fixed points.

翻译：我们证明，对于一个性质良好的函子 $F$，相对初始或相对终结不动点构成 $F$-余代数与 $F$-代数之间的一对伴随函子。我们利用局部可表现范畴的语言，寻找该伴随存在性的充分条件。我们证明，相对不动点可刻画为 $F$ 上自由（余）单子的（余）等化子。特别地，当 $F$ 是 $\mathsf{Set}$ 上的多项式函子时，相对不动点是自由项代数或余自由项余代数的商或子集。我们给出多项式函子相对不动点的例子，以及一个谢尔宾斯基地毯的例子。最后，我们证明关于相对不动点的一个一般保持性结果。