A name for 'lax' colimit for poset-valued functors? Let $F,G\colon C\to \mathsf{POS}$ be functors, where $C$ is a category and $\mathsf{POS}$ denotes the category of posets and monotone maps.
By a lax natural transformation $\alpha\colon F\Rightarrow G\colon C\to\mathsf{POS}$ I mean an $\mathsf{Ob}(C)$-indexed family $(F(c)\xrightarrow{\alpha_c}G(c))_c$ of monotone maps such that for each $f\in C(c,d)$, $G(f)\circ\alpha_c\leq \alpha_d\circ F(f)$, where $\leq$ stands for the point-wise order relation (induced by $G(d)$).
Let $(X,\leq)$ be a poset. By a lax cone with base $F$ to $(X,\leq)$ is meant a lax natural transformation $\tau\colon F\Rightarrow \Delta(X,\leq)$, where $\Delta(X,\leq)\colon C\to \mathsf{POS}$ is the usual diagonal functor with constant value $(X,\leq)$.
Finally I call lax colimit of $F$ a pair $((X_F,\leq_F),q^F)$, where $(X_F,\leq_F)$ is a poset and $q^F\colon F\Rightarrow \Delta(X_F,\leq_F)$ is a lax cone with base $F$ to $(X_F,\leq_F)$ such that for any other lax cone $\tau\colon F\Rightarrow \Delta(X_F,\leq_F)$, there is a unique monotone map $t\colon (X_F,\leq_F)\to (X,\leq)$ such that for each $C$-object $c$, $t\circ q_c^F=\tau_c$.
In other words the definition of the lax colimit for a poset-valued functor is the same as the definition of its usual colimit except that ordinary cones are replaced by lax ones (that is, equality is replaced by inequality).
I imagine that there is a standard terminology for this kind of notions. But can someone tell me?
 A: $\textbf{Pos}$ is a slightly degenerate example of a 2-category, where there is at most one 2-morphism between any parallel pair of 1-morphisms: explicitly, given $f, g : X \to Y$, we have a 2-morphism $f \Rightarrow g$ if and only if $f (x) \le g (x)$ for all $x \in X$.
Thus, we may apply notions from 2-category theory to $\textbf{Pos}$.
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$ and 2-functors $F, G : \mathcal{C} \to \mathcal{D}$, a lax natural transformation $\alpha : F \Rightarrow G$ is an assignment of a 1-morphism $\alpha_X : F X \to G X$ for each object $X$ in $\mathcal{C}$ and a 2-morphism $\alpha_f : G f \circ \alpha_X \Rightarrow \alpha_Y \circ F f$ for each 1-morphism $f : X \to Y$ in $\mathcal{C}$, such that for every object $X$ in $\mathcal{C}$, $\alpha_{\textrm{id}_X} = \textrm{id}_{\alpha_X}$, and for every composable pair $f : X \to Y$ and $g : Y \to Z$ in $\mathcal{C}$, $\alpha_g G f \bullet F g \alpha_f = \alpha_{g \circ f}$, i.e. $\alpha_{g \circ f}$ is equal to the following composite 2-morphism:
$$\require{AMScd}
\begin{CD}
F X @>{\alpha_X}>> G X \\
@V{F f}VV \overset{\alpha_f}{\Leftarrow} @VV{G f}V \\
F Y @>{\alpha_Y}>> G Y \\
@V{F g}VV \overset{\alpha_g}{\Leftarrow} @VV{G g}V \\
F Z @>>{\alpha_Z}> G Z
\end{CD}$$
In the special case of $\textbf{Pos}$ this agrees with what you have defined.
A lax limit of a 2-functor $F : \mathcal{C} \to \mathcal{D}$ weighted by a 2-functor $J : \mathcal{C} \to \textbf{Cat}$ is an object $L$ in $\mathcal{D}$ equipped with a universal lax natural transformation $J \Rightarrow \mathcal{D} (L, F {-})$, i.e. a representation of the 2-functor $\textbf{Lax} (J, \mathcal{D} ({-}, F))$.
Specialising to the case where $J$ is the constant $1$, we get the notion of a lax conical limit of $F$.
A bit more explicitly, a lax cone over $F : \mathcal{C} \to \mathcal{D}$ is an assignment of a 1-morphism $\lambda_X : L \to F X$ for each object $X$ in $\mathcal{C}$ and a 2-morphism $\lambda_f : F f \circ \lambda_X \Rightarrow \lambda_Y$ for each 1-morphism $f : X \to Y$ in $\mathcal{C}$, such that for every object $X$ in $\mathcal{C}$, $\lambda_{\textrm{id}_X} = \textrm{id}_{\lambda_X}$, and for every composable pair $f : X \to Y$ and $g : Y \to Z$ in $\mathcal{C}$, $\lambda_g F f \bullet \lambda_f = \lambda_{g \circ f}$, i.e. $\lambda_{g \circ f}$ is equal to the following composite 2-morphism:
$$\begin{CD}
L @>{\lambda_X}>> F X \\
@| \overset{\lambda_f}{\Leftarrow} @VV{F f}V \\
L @>{\lambda_Y}>> F Y \\
@| \overset{\lambda_g}{\Leftarrow} @VV{F g}V \\
L @>>{\lambda_Z}> F Z
\end{CD}$$
A lax conical limit of $F$ is then an object equipped with a universal lax cone over $F$.
Where things get tricky is where you dualise.
A lax colimit of a 2-functor $F : \mathcal{C} \to \mathcal{D}$ weighted by a 2-functor $J : \mathcal{C}^\textrm{op} \to \textbf{Cat}$ is an object $L$ in $\mathcal{D}$ equipped with a universal lax natural transformation $\lambda : J \Rightarrow \mathcal{D} (F {-}, L)$.
Unfolding, that means we have a functor $\lambda_X : J X \to \mathcal{D} (F X, L)$ for each object $X$ in $\mathcal{C}$ and a natural transformation $\lambda_f : \mathcal{D} (F f, L) \circ \lambda_Y \Rightarrow \lambda_X \circ J f$ for each 1-morphism $f : X \to Y$ in $\mathcal{C}$, satisfying various equations.
Specialising, we find that a lax cocone under $F$ is an assignment of a 1-morphism $\lambda_X : F X \to L$ for each object $X$ in $\mathcal{C}$ and a 2-morphism $\lambda_f : \lambda_Y \circ F f \Rightarrow \lambda_X$ for each 1-morphism $f : X \to Y$ in $\mathcal{C}$ such that etc.
This is not the same thing as a lax natural transformation from $F$ to the constant $L$!
Instead, a lax natural transformation from $F$ to a constant is an oplax cocone under $F$.
Thus, what you have defined is the notion of an oplax conical colimit.
