Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.

A well known theorem independently discovered by Eilenberg and Watts states that every $k$-linear, right exact, cocontinuous functor $F: A\mathsf{-Mod}\to B\mathsf{-Mod}$ is of the form $M\otimes_A-$ for some bimodule ${}_B M_A$ (in fact $M=F(A)$).

A theorem of Rickard (with some refinements by other people like Keller) also states that if there is an equivalence $D^b(A) \to D^b(B)$ there is also an equivalence of the form $X\otimes_A^\mathbb{L}-$ for some $X\in D^b(B\otimes_k A^{op})$.

A reasonable question to ask is then:

Is every equivalence $D^b(A) \to D^b(B)$ itself of the form $X\otimes_A^{\mathbb{L}}-$ for some bounded complex of bimodules $X$?

or more generally:

Is every exact (plus maybe some other reasonable condition) functor $D^b(A) \to D^b(B)$ of triangulated categories of the form $X\otimes_A^\mathbb{L}-$ for some bounded complex of bimodules $X$ ?

Let $A$ and $B$ be $k$-algebras. And for convenience let's say $k$ is a field and both $A$ and $B$ are finite-dimensional.

A well known theorem independently discovered by Eilenberg and Watts states that every $k$-linear, right exact, cocontinuous functor $F: A\mathsf{-Mod}\to B\mathsf{-Mod}$ is of the form $M\otimes_A-$ for some bimodule ${}_B M_A$ (in fact $M=F(A)$).

A theorem of Rickard (with some refinements by other people like Keller) also states that if there is an equivalence $D^b(A) \to D^b(B)$ there is also an equivalence of the form $X\otimes_A^\mathbb{L}-$ for some $X\in D^b(B\otimes_k A^{op})$.

A reasonable question to ask is then:

Is every equivalence $D^b(A) \to D^b(B)$ itself of the form $X\otimes_A^{\mathbb{L}}-$ for some bounded complex of bimodules $X$?

or more generally:

Is every exact functor $D^b(A) \to D^b(B)$ of triangulated categories which commutes with all direct sums (plus maybe some other reasonable condition) of the form $X\otimes_A^\mathbb{L}-$ for some bounded complex of bimodules $X$ ?