Suppose we are given complete lattices $(L_i,\le_i)$ indexed over some set $I$, such that $L_i\ne 2$ and $L_i\cap L_j=\emptyset$ for all $i\ne j$. There is a sort of "amalgamation" that can be constructed from these lattices. Consider $L$ as the union of the sets $L_i\setminus\{0_L,1_L\}$ alongisde two objects $0,1\not\in\bigcup_{i\in I} L_i$. Construct a partial order as follows: $0\le x \le 1$ for all $x\in L$, and $x\le y$ if $x,y\in L_i\setminus\{0_L,1_L\}$ for some $i$ and $x\le_i y$. This makes $(L,\le)$ a complete lattice which glues together all of the original lattices by identifying the least and greatest elements.
Does this construction have a name? I have searched for it all over the internet and books but cannot seem to find anything about it. Maybe it is a particular case of a different lattice construction?
I have seen this is called the Horizontal Sum of the bounded lattices $(L_i, \land_i, \lor_i, 0_i, 1_i)$.
As is often the case in order theory, the term "Horizontal Sum" means something different depending on how much structure you have. For mere posets $(P_i, \leq_i)$, the horizontal sum is defined as
$$x_i \leq y_j \iff i = j \text{ and } x_i \leq_i y_j$$
(so we place the posets next to each other horizontally). Davey and Priestley call this the "disjoint union" of two posets, denoted $\overset{\cdot}{\cup}$.
However, when our posets are bounded, that is, when there is a $0_i, 1_i \in P_i$ with $0_i \leq x \leq 1_i$ for each $x \in P_i$, then we can ask for all of our poset operations to respect these bounds. In this case we write $(P_i, \leq_i, 0_i, 1_i)$ to indicate that there is extra structure we're keeping track of.
In this case, a "Horizontal Sum" is the construction you've described: We take the horizontal sum as mere posets, but then identify all the top (resp. bottom) elements.
I'm struggling to find a textbook reference for this, but you can see this definition in Chajda and Länger's Horizontal Sums of Bounded Lattices or implicit in Giuntini, Mureşan, and Paoli's Ordinal and Horizontal Sums Constructing PBZ*-lattices.
I've checked my lattice theory references, but I'm not finding this particular definition anywhere (even though I know I've seen it). I'll look through my combinatorics references too to see if it's in one of those, and I'll update this answer if I'm able to find a textbook reference.
Edit: Aha! I thought it was in Davey and Priestley's Introduction to Lattices and Order. It just wasn't where I expected. You can find this definition relegated to exercise $3.12$, on page $84$ of my edition.
I hope this helps ^_^