Let me give a simple necessary and sufficient condition which characterizes which directed sets $D$ satisfy the property that $Hom(P,D)$ is directed for all posets $P$.

Suppose that $D$ is a poset. Then I claim that $Hom(P,D)$ is directed for every poset $P$ if and only if there is a function $L:D\times D\rightarrow D$ such that $L(x,y)\geq x,L(x,y)\geq y$ for each $x,y\in D$ and where if $x_{1}\leq x_{2},y_{1}\leq y_{2}$, then $L(x_{1},y_{1})\leq L(x_{2},y_{2})$. In other words, $Hom(P,D)$ is directed for each poset $P$ if and only if $D$ is $``$monotonely directed$"$ or
$``$uniformly directed$"$. The proof is straightforward, but let me give a proof below regardless.

Suppose there is such a function $L:D\times D\rightarrow D$. Then whenever $f,g\in Hom(P,D)$ are order preserving, then define $h(x)=L(f(x),g(x))$. Then whenever $x\leq y$, we have $f(x)\leq f(y),g(x)\leq g(y)$, so $h(x)=L(f(x),g(x))\leq L(f(y),g(y))=h(y)$, so $h$ is order preserving. Furthermore, $h(x)=L(f(x),g(x))\geq f(x)$ and $h(x)=L(f(x),g(x))\geq g(x)$, so $h\geq f$ and $h\geq g$, so $Hom(P,D)$ is always directed.

Now suppose that $Hom(P,D)$ is directed for each poset $P$. Then $Hom(D\times D,D)$ is directed where $D\times D$ is given the partial ordering where $(x_{1},y_{1})\leq(x_{2},y_{2})$ if and only if $x_{1}\leq x_{2}$ and $y_{1}\leq y_{2}$. Let $\pi_{1},\pi_{2}:D\times D\rightarrow D$ be the projection mappings. In other words, $\pi_{1}(x,y)=x,\pi_{2}(x,y)=y$. Then $\pi_{1},\pi_{2}$ are order preserving, so there is some $L\in Hom(D\times D,D)$ with $\pi_{1},\pi_{2}\leq L$. Therefore, $L$ is an order preserving map with $L(x,y)\geq\pi_{1}(x,y)=x$ and $L(x,y)\geq\pi_{2}(x,y)=y$.