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Starting from this code

documentclass[12pt]book
usepackagemathtools,amssymb 
usepackagenewtxtext
usepackage[lite]mtpro2
begindocument
$(Lboldsymbolmathcal A')^intercal =boldsymbolmathcal A'^intercal L^intercal $
enddocument

with this output

I'd like to move the intercal command a little higher and to the left in the direction indicated by the arrow. But does LaTeX have a direct command to move the symbol up or down or do I need to use specific macros? What could be the ways to have an adequate and beautiful way to write this formula?

Combining David's suggestion with Sebastiano's requirement...

I use mathsfT as the core of the revised intercal (you can rename it something else if you don't want to overwrite it), but then to make it smaller, I use it in cramped, superscripted form (revision 1). Alternately, I just set it in scriptscriptstyle, which lowers its elevation a bit, while still keeping it higher than the original intercal (revision 2).

Original, then two revisions.

documentclass[12pt]book
usepackagemathtools,amssymb 
usepackagenewtxtext
usepackagenewtxmath%[lite]mtpro2
begindocument
$(Lboldsymbolmathcal A')^intercal =boldsymbolmathcal A'^intercal L^intercal $

renewcommandintercalcramped^mathsfT
$(Lboldsymbolmathcal A')^intercal =boldsymbolmathcal A'^intercal L^intercal $

renewcommandintercalmathsfscriptscriptstyle T
$(Lboldsymbolmathcal A')^intercal =boldsymbolmathcal A'^intercal L^intercal $
enddocument

Here's another variation on David's idea to use mathsfT instead of either intercal or top. The macro tp (short for "transpose", naturally) employs scriptscriptstyle math mode. The tp macro takes an optional argument: a scalar number, to indicate how many mu the "T" should be shifted to the left. The ability to fine-tune the horizontal position can come in handy if you're dealing with letters, such as L, which have no component in the upper-right quadrant. (The default amount of left-shift is 3, as in -3mu. Aside: mkern-3mu is the same as !.)

documentclass[12pt]book
usepackagemathtools,amssymb,newtxtext
usepackage[lite]mtpro2
newcommandbmcAboldsymbolmathcal A'
newcommandtp[1][3]^^mkern-#1mumathsfT
begindocument
$intercal$ $top$ $mathsfT$

medskip
begintabular@ll
$(LbmcA)^intercal = bmcA^intercal L^intercal$ & verb+intercal+ \
$(LbmcA)^top = bmcA^top L^top$ & verb+top+ \
$(LbmcA)tp = bmcAtp Ltp = bmcAtp Ltp[6]$ & verb+tp+
endtabular
enddocument