Resolving ambiguity around Lattice Enthalpy in Born-Haber Cycle

According to my textbook,



> 
> Lattice enthalpy is the enthalpy change that occurs when one mole of a solid ionic compound is separated into gaseous ions under standard conditions.$^1$
> 
> 
> 


According to the same textbook, this is an endothermic process, which makes sense as a lattice is generally held together by strong ionic bonds and thus would require energy to separate the atoms. The following diagram is given for the born-haber cycle in the textbook:


[![enter image description here](https://i.stack.imgur.com/ZcL2e.png)](https://i.stack.imgur.com/ZcL2e.png)


The diagram supports the definition as the enthalpy of the individual gaseous atoms is greater than that of the lattice, i.e. the arrow moves up. **However**, as I was working through past papers by the International Baccalaureate, a question asked to draw the born-haber cycle for LiF.$^2$ This was the answer:


[![enter image description here](https://i.stack.imgur.com/f5r8Q.png)](https://i.stack.imgur.com/f5r8Q.png)


In this case, the arrow is pointing down for the lattice enthalpy. To me, this does not suite the definition of lattice enthalpy as separating a lattice into its ions would require energy and not release energy. Hence, should the arrow not be pointing upwards for $\ce{LiF\_{(s)}}$ to $\ce{Li^+\_{(g)} + Cl^-\_{(g)}}$? Am I making a fundamental error, is the case for NaCl different to LiF or is the answer wrong?


Works Cited:


1. Pearson Baccalaureate: Higher Level Chemistry 2nd Edition. By Catrin Brown and Mike Ford
2. International Baccalaureate


